\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)
Let \(A \in M_n(\mathbb{K})\) be an upper triangular matrix with all diagonal coefficients equal to \(1\) (i.e.\ \(a_{ii} = 1\) for every \(i\)). Show that \(A\) is invertible and that \(A^{-1}\) is also upper triangular with all diagonal coefficients equal to \(1\).
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